Non-injectivity of the \(q\)-deformed von Neumann algebra. (English) Zbl 1060.46051

Nou, Alexandre

Math. Ann. 330, No. 1, 17-38 (2004).

Summary: We prove that the von Neumann algebra generated by \(q\)-Gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a suitable vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called \(q\)-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non-injectivity under some assumption on the spectral set of the positive operator associated with the deformation.

MSC:

46L65	Quantizations, deformations for selfadjoint operator algebras
46L54	Free probability and free operator algebras